Chapter 12: Area of Shapes

12.1: Area of Rectangles

Area of Rectangles

•The Area of an L-unit by W-unit rectangle is• Area = L x W

•True for any non-negative values L and W

Section 12.2: Moving and Additive Principles

About Area

The Moving and Additive Principles•Moving Principle: If you move a shape rigidly without stretching it, then its area does not change.• Rigid motions include translations, reflections, and rotations

•Additive Principle: If you combine a finite number of shapes without overlapping them, then the area of the resulting shape is the sum of the areas of the individual shapes.

Example Problems•Ex 1: Determine the area of the following shape.

Ex 1:

•Ex 2: Determine the • area of the following • shape.

•Ex 3: The UK Math Department is going to retile hallway of the 7th floor of POT, shown below. How many square feet is the hallway?

See Activity 12 C, problem 1

•Ex 4: Determine the area of the following hexagon.

Section 12.3: Area of Triangles

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Example Problem• Ex 1: Determine the area of

the following triangle.

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Triangle Definitions• Def: The base of a triangle is any of its three sides

• Def: Once the base is selected, the height is the line segment that• is perpendicular to the base &• connects the base or an extension of it to the opposite vertex

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Base and Height Ex’s

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Area of a Triangle

• The area of a triangle with base b and height h is given by the formula

• It doesn’t matter which side you choose as the base!

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Revisiting Example 1• Ex 1: Determine the area of

the following triangle.

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•See problems in Activities 12F and 12G

Section 12.4: Area of

Parallelograms and Other Polygons

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•See Activity 12H

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Definitions for Parallelograms• Def: The base of a parallelogram is any of its four sides

• Def: Once the base is selected, the height of a parallelogram is a line segment that• perpendicular to the base &• connects the base or an extension of it to a vertex on not on

the base

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Area of a Parallelogram

• The area of a parallelogram with base b and height h is

Section 12.5: Shearing

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What is shearing?

Def: The process of shearing a polygon:• Pick a side as its base• Slice the polygon into infinitesimally thin strips that are parallel

to the base• Slide strips so that they all remain parallel to and stay the

same distance from the base

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Examples of Shearing

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Result of Shearing• Cavalieri’s Principle: The original and sheared shapes

have the same area.

Key observations during the shearing process:• Each point moves along a line parallel to the base• The strips remain the same length• The height of the stacked strips remains the same

Section 12.6: Area of Circles and the Number π

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Definitions• Def: The circumference of a circle is the distance around

a circle

• Recall: radius- the distance from the center to any point on

the circle diameter- the distance across the circle through

the center

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The number π• Def: The number pi, or π, is the ratio of the circumference and diameter of any circle. That is, • Circumference Formulas: The circumference of a circle is

given by or

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Quick Example Problem

• Ex 1: A circular racetrack with a radius of 4 miles has what length for each lap?

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How to demonstrate the size of π• See activities 12M and 12N

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Area of a Circle• The area of a circle with radius is given by

• See Activity 12O to see why

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Example Problems• Ex 2: If you make a 5 foot wide path around a circular

courtyard that has a 15 foot radius, what is the area of the new path?

• Ex 3: A mile running track has the following shape consisting of a rectangle with 2 semicircles on the ends. If you are planting sod inside the track, how many square feet of sod do you need?

Section 12.7: Approximating Areas of Irregular Shapes

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How do we estimate the area of the following shape?

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Methods for Estimating Area• Graph Paper:

1. Draw/trace shape onto graph paper2. Count the approximate number of squares inside the shape3. Convert the number of squares into a standard unit of area based on the

size of each square• Modeling Dough:

1. Cover the shape with a layer (of uniform thickness) of modeling dough2. Reform the dough into a regular shape such as a rectangle or circle (of the

same thickness)3. Calculate the area of the regular shape

• Card Stock:1. Draw/trace shape onto card stock2. Cut out the shape and measure its weight3. Weigh a single sheet of card stock4. Use ratios of the weights and the area of one sheet to estimate the

shape’s area

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See Example problems in Activity 12Q